Required tools

Making a round hole

In this mini tutorial we are going to begin with a cube, but you can apply this technique to make round holes to any mesh, whatever its shape.

We are going to make a round hole that goes through the cube, so the fist step is to select two opposite faces.

In Edit mode select two opposite faces

Select the menu option Mesh -> Edges -> Subdivide

Select again the menu option Mesh -> Edges -> Subdivide. Now each of the two faces you selected has been subdivided into 16 smaller faces as shown in picture Set the cursor at the centre of the face.

Select the central vertex of one of the faces and select the menu option Mesh -> Snap -> "Cursor -> Selection" as shown in picture Set the cursor at the centre of the face. Now the cursor is at the centre of that face.

Select the 8 vertices that are around the central vertex (don't select the central vertex) and click on the To Sphere button on the Mesh Tools area (see picture Move vertices to form a circle). Click OK to accept the 100% Percentage value on the pop up window. You'll see that the 8 vertices you selected have moved outlining a circle.

Delete the central vertex: select it, press the Delete key and select Vertices from the pop up window. Now you have a round hole on one of the faces of the cube as shown in picture Round hole on one face.

Repeat the process from 4. to 6. on the opposite face of the cube to make another round hole on that face.

Now we are going to create interior faces to join the two holes:

Select 4 vertices: two of the top face and two of the bottom face, and select the menu option Mesh -> Make Edge/Face as shown in picture Make interior faces. This will create a face that has each of those 4 vertices as corners.

Repeat the process selecting the next 4 vertices and creating a face, and go on until having created all 8 interior faces as shown in picture 8 interior faces

Go to Object mode and add the Bevel modifier (click on Add Modifier and select Bevel from the drop down list) to smooth the circle as shown in picture Round hole and you'll be done.